Types of Relation in Maths

A relation is like a rule that tells us how two things are connected. For example:

• In a classroom, the relation can be "is the friend of."
• In numbers, the relation can be "is greater than" or "is equal to."

Some of the common types of relations in maths are:

Let's discuss them in detail.

Universal Relation

A relation R on a set A is called a universal relation if every element in A × A is related.

Example: For set A = {1, 2}, R = A × A = {(1, 1), (1, 2), (2, 1), (2, 2)}.

Empty Relation

A relation R is empty if it contains no ordered pairs.

Example: For A = {1, 2}, R = ∅.

Identity Relation

A relation R on set A is called an identity relation if every element is related to itself and no other elements.

Example: For A = {1, 2, 3}, R = {(1, 1), (2, 2), (3, 3)}.

Reflexive Relation

R is reflexive if (a, a) ∈ R for all a ∈ A.

Example: For A = {1, 2}, R = {(1, 1), (2, 2), (1, 2)}.

Symmetric Relation

R is symmetric if (a, b) ∈ R ⇒ (b, a) ∈ R.

Example: If R = {(1, 2), (2, 1)}, R is symmetric.

Anti-Symmetric Relation

R is anti-symmetric if (a, b) ∈ R and (b, a) ∈ R ⇒ a = b.

Example: If R = {(1, 2)}, R is anti-symmetric.

Transitive Relation

R is transitive if (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R.

Example: If R = {(1, 2), (2, 3), (1, 3)}, R is transitive.

Equivalence Relation

R is an equivalence relation if it is reflexive, symmetric, and transitive.

Example: Relation R defined by "is of the same age group" is an equivalence relation.

Partial Order Relation

R is a partial order if it is reflexive, anti-symmetric, and transitive.

Example: ≤ on the set of integers.

Total Order Relation

R is a total order if it is a partial order and every pair of elements is comparable.

Example: ≤ on R.

Inverse Relation

The inverse of a relation R is defined as R⁻¹ = {(b, a) : (a, b) ∈ R}.

Example: If R = {(1, 2), (3, 4)}, then R⁻¹ = {(2, 1), (4, 3)}.

Complement of a Relation

The complement of R is defined as R^c = (A × A) − R.

Example: For A = {1, 2}, R = {(1, 1)}, R^c = {(1, 2), (2, 1), (2, 2)}.

Void Relation

A relation with no pairs, R = ∅, is called a void relation.

Example: For any set A, R = ∅ is void.

Other Types of Relations

Apart from the property-based relations like reflexive or symmetric, relations can also be classified according to how elements from one set are mapped to elements of another set.

1. One-to-One Relation

A relation is one-to-one if each element of the first set is paired with a unique element of the second set, and no two elements from the first set share the same image.

Example:

Let A = {1, 2, 3} and B = {4, 5, 6}
R = {(1, 4), (2, 5), (3, 6)} is one-to-one.

2. One-to-Many Relation

A relation is one-to-many if a single element of the first set is paired with more than one element of the second set.

Example:

Let A = {1} and B = {4, 5, 6},
R = {(1, 4), (1, 5), (1, 6)} is one-to-many.

3. Many-to-One Relation

A relation is many-to-one if two or more different elements of the first set are paired with the same element of the second set.

Example:

Let A = {1, 2, 3} and B = {4},
R = {(1, 4), (2, 4), (3, 4)} is many-to-one.

4. Many-to-Many Relation

A relation is many-to-many if one or more elements of the first set are paired with multiple elements of the second set.

Example:

Let A = {1, 2} and B = {4, 5},
R = {(1, 4), (1, 5), (2, 4), (2, 5)} is many-to-many.

Also Check

• Relation and Function
• Types of Function